Persistence and stationary distribution of a stochastic predator-prey model under regime switching

Li Zu; Daqing Jiang; Donal O'Regan

doi:10.3934/dcds.2017124

Article Contents

2017, Volume 37, Issue 5: 2881-2897. Doi: 10.3934/dcds.2017124

This issue Previous Article On the limit quasi-shadowing property Next Article

Persistence and stationary distribution of a stochastic predator-prey model under regime switching

1.
College of Mathematics and Statistics, Hainan Normal University, Hainan 571158, China
2.
School of Science, China University of Petroleum (East China), Qingdao 266580, China
3.
Nonlinear Analysis and Applied Mathematics(NAAM)-Research Group, King Abdulaziz University, Jeddah, Saudi Arabia
4.
School of Mathematics, Statistics and Applied Mathematics, National University of Ireland, Galway, Ireland

* Corresponding author: Daqing Jiang (School of Science, China University of Petroleum (East China), Qingdao), China E-mail: daqingjiang2010@hotmail.com, Tel.: +86 43185099589; fax: +86 43185098237
* Corresponding author: Daqing Jiang (School of Science, China University of Petroleum (East China), Qingdao), China E-mail: daqingjiang2010@hotmail.com, Tel.: +86 43185099589; fax: +86 43185098237

Received: October 2014

Revised: January 2017

Published: May 2017

The first author is supported by NSFC of China (No: 11601038), the second author is supported by NSFC of China (No: 11371085) and the Fundamental Research Funds for the Central Universities (No: 15CX08011A).

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Abstract

Taking both white noise and colored environment noise into account, a predator-prey model is proposed. In this paper, our main aim is to study the stationary distribution of the solution and obtain the threshold between persistence in mean and the extinction of the stochastic system with regime switching. Some simulation figures are presented to support the analytical findings.

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Mathematics Subject Classification: Primary: 34F05, 92D25; Secondary: 92Bxx.

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Figure 1. The curves on the subgraphs (a) and (b) are the density functions of x(t) and y(t) in k = 1, k = 2, respectively. The white noise intensity on x(t) and y(t) are all relatively small.

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Figure 2. The subgraphs (a) and (b) denote sample phase portrait of stochastic system and the corresponding deterministic system. The red, blue and black area denote x(t), y(t) in k = 1, k = 2 and converting between the above two states, respectively. The white noise intensity on x(t) and y(t) are all relatively small.

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Figure 3. The subgraphs (a) and (b) have the same definitions as in Fig.2. The white noise intensity on x(t) and y(t) are all relatively big.

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